To demonstrate the concept of dividing polynomials using the inverse relationship of multiplication, we want to rearrange the division problem into a multiplication equation.
The expression we are working with is:
\[ \frac{8x^2 - 4x + 12}{4x} \]
We want to express it in the form \( A = (B)(C) \), where \( A \) is the original expression, \( B \) is the divisor, and \( C \) is the quotient.
None of the options you provided seem to match perfectly with a standard interpretation of the division as multiplication. However, the concept being illustrated is that you can express the division as a product where one factor is the reciprocal of the divisor.
So, if we consider:
\[ \frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right) (8x^2 - 4x + 12) \]
This would correspond to a choice where:
\[ \frac{8x^2 - 4x + 12}{4x} = \left(\frac{1}{4x}\right)(8x^2 - 4x + 12) \]
Given your options, the one that correctly demonstrates this relationship is:
Start Fraction 8 x squared minus 4 x plus 12 over 4 x End Fraction equals left parenthesis negative Start Fraction 1 over 4 x End Fraction right parenthesis left parenthesis 8 x squared minus 4 x plus 12 right parenthesis
Which can be simplified further if needed to express how dividing by \( 4x \) can be thought of as multiplying by its reciprocal.